International Journal of Game Theory

In earlier work, the authors showed that a pure-strategy Bayesian-Nash equilibria in games with uncountable action sets and atomless private information spaces may not exist if the information space of each player is not saturated. This paper sharpens this result by exhibiting a failure of the existence claim for a game in which the information space of only one player is not saturated. The methodology that enables this extension of the necessity theory is novel relative to earlier work, and its conceptual underpinnings may have independent interest.

We consider a demand bargaining model where the formateur (proposal maker) is appointed before the players state their demands. In a general continuous framework, we show that if the decision has a distributive dimension, then the formateur gets his way.

In this paper, we address various types of two-person stochastic games—both zero-sum and nonzero-sum, discounted and undiscounted. In particular, we address different aspects of stochastic games, namely: (1) When is a two-person stochastic game completely mixed? (2) Can we identify classes of undiscounted zero-sum stochastic games that have stationary optimal strategies? (3) When does a two-person stochastic game possess symmetric optimal/equilibrium strategies? Firstly, we provide some necessary and some sufficient conditions under which certain classes of discounted and undiscounted stochastic games are completely mixed. In particular, we show that, if a discounted zero-sum switching control stochastic game with symmetric payoff matrices has a completely mixed stationary optimal strategy, then the stochastic game is completely mixed if and only if the matrix games restricted to states are all completely mixed. Secondly, we identify certain classes of undiscounted zero-sum stochastic games that have stationary optima under specific conditions for individual payoff matrices and transition probabilities. Thirdly, we provide sufficient conditions for discounted as well as certain classes of undiscounted stochastic games to have symmetric optimal/equilibrium strategies—namely, transitions are symmetric and the payoff matrices of one player are the transpose of those of the other. We also provide a sufficient condition for the stochastic game to have a symmetric pure strategy equilibrium. We also provide examples to show the sharpness of our results.

The modiclus, a relative of the prenucleolus, assigns a singleton to any cooperative TU game. We show that the modiclus selects a member of the core for any exact orthogonal game and for any assignment game that has a stable core. Moreover, by means of an example we show that there is an exact TU game with a stable core that does not contain the modiclus.

LetX denote a linear space of real valued functions defined on a subset of a Banach space such thatX containsE′ the dual space ofE as a subspace. Given a distinguished vectorx 0 inE anx 0-value (onX) is defined to be a projectionP fromX ontoE′ which satisfies the following two hypotheses: (VA) (PF)(x0)=Fx0 for allF inX; (VB) IfT is a continuous isomorphism fromE intoE such thatTx 0=x 0 thenP(F⋄T) = (PF) ⋄ T for allF inX. The existence and uniqueness of a value is established for two choices ofX, one of which is the space of polynomials in functional onE. The existence and partial uniqueness of a value is established on a third choice forX.

In the classical secretary problem the decision maker can only observe the relative ranks of the items presented. Recently, Ferguson — building on ideas of Stewart — showed that, in a game theoretic sense, there is no advantage if the actual values of the random variables underlying the relative ranks can be observed (game of googol). We extend this to the case where the number of items is unknown with a known upper bound. Corollary 3 extends one of the main results in [HK] toall randomized stopping times. We also include a modified, somewhat more formal argument for Ferguson's result.

The main goal of this paper is to present a different perspective than the more ‘traditional’ approaches to study solutions for games with externalities. We provide a direct sum decomposition for the vector space of these games and use the basic representation theory of the symmetric group to study linear symmetric solutions. In our analysis we identify all irreducible subspaces that are relevant to the study of linear symmetric solutions and we then use such decomposition to derive some applications involving characterizations of classes of solutions.